Branching rules for level-zero extremal weight modules from $U_q(\widehat{\mathfrak{sl}}_{n+1})$ to $U_q(\widehat{\mathfrak{sl}}_n)$

TL;DR

This paper investigates how level-zero extremal weight modules over quantum affine algebras decompose when restricted from $U_q(\uppi ext{-sl}_{n+1})$ to $U_q(\uppi ext{-sl}_n)$, revealing explicit direct sum structures and isomorphisms.

Contribution

It provides a detailed decomposition of modules under an algebra homomorphism, extending understanding of module structures in quantum affine algebras.

Findings

Established a direct sum decomposition of the modules.

Identified isomorphisms with tensor products of extremal weight modules and Laurent polynomial rings.

Proved that for certain weights, modules decompose into sums of extremal weight modules.

Abstract

In this paper, we study the structure of a $U_q(\widehat{\mathfrak{sl}}_n)$-module $\Psi_{\varepsilon}^* V(\lambda)$, where $V(\lambda)$ is the extremal weight module of level-zero dominant weight $\lambda$ over the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_{n+1})$ and $\Psi_{\varepsilon}: U_q(\widehat{\mathfrak{sl}}_n) \to U_q(\widehat{\mathfrak{sl}}_{n+1})$ is an injective algebra homomorphism. We establish a direct sum decomposition $\Psi_{\varepsilon}^* V(\lambda) \cong M_{0,\varepsilon} \oplus \cdots \oplus M_{m,\varepsilon}$, where $M_{0,\varepsilon}$ and $M_{m,\varepsilon}$ are isomorphic to a tensor product of an extremal weight module over $U_q(\widehat{\mathfrak{sl}}_n)$ and a symmetric Laurent polynomial ring. Moreover, when $\lambda$ is a multiple of a level-zero fundamental weight, we show that $\Psi_{\varepsilon}^* V(\lambda)$ is isomorphic to a direct sum of extremal weight modules.